Multilinear Function

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

:$f\colon V_1 \times \cdots \times V_n \to W\text$

where $V_1,\ldots,V_n$ and $W$ are vector spaces (or modules over a commutative ring), with the following property: for each $i$, if all of the variables but $v_i$ are held constant, then $f(v_1, \ldots, v_i, \ldots, v_n)$ is a linear function of $v_i$.

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.

If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating ''k''-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.

Examples

* Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in $\mathbb^3$.
* The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
* If $F\colon \mathbb^m \to \mathbb^n$ is a ''C^k'' function, then the $k\!$th derivative of $F\!$ at each point $p$ in its domain can be viewed as a symmetric $k$-linear function $D^k\!F\colon \mathbb^m\times\cdots\times\mathbb^m \to \mathbb^n$.

Coordinate representation

Let

:$f\colon V_1 \times \cdots \times V_n \to W\text$

be a multilinear map between finite-dimensional vector spaces, where $V_i\!$ has dimension $d_i\!$, and $W\!$ has dimension $d\!$. If we choose a basis $\$ for each $V_i\!$ and a basis $\$ for $W\!$ (using bold for vectors), then we can define a collection of scalars $A_^k$ by

:$f(\textbf_,\ldots,\textbf_) = A_^1\,\textbf_1 + \cdots + A_^d\,\textbf_d.$

Then the scalars $\$ completely determine the multilinear function $f\!$. In particular, if

:$\textbf_i = \sum_^ v_ \textbf_\!$

for $1 \leq i \leq n\!$, then

:$f(\textbf_1,\ldots,\textbf_n) = \sum_^ \cdots \sum_^ \sum_^ A_^k v_\cdots v_ \textbf_k.$

Example

Let's take a trilinear function

:$g\colon R^2 \times R^2 \times R^2 \to R,$
where , and .

A basis for each is $\ = \ = \.$ Let

:$g(\textbf_,\textbf_,\textbf_) = f(\textbf_,\textbf_,\textbf_) = A_,$
where $i,j,k \in \$. In other words, the constant $A_$ is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three $V_i$), namely:
:$\, \, \, \, \, \, \, \.$

Each vector $\textbf_i \in V_i = R^2$ can be expressed as a linear combination of the basis vectors

:$\textbf_i = \sum_^ v_ \textbf_ = v_ \times \textbf_1 + v_ \times \textbf_2 = v_ \times (1, 0) + v_ \times (0, 1).$

The function value at an arbitrary collection of three vectors $\textbf_i \in R^2$ can be expressed as
:$g(\textbf_1,\textbf_2, \textbf_3) = \sum_^ \sum_^ \sum_^ A_ v_ v_ v_.$
Or, in expanded form as
:$\begin g((a,b),(c,d)&, (e,f)) = ace \times g(\textbf_1, \textbf_1, \textbf_1) + acf \times g(\textbf_1, \textbf_1, \textbf_2) \\ &+ ade \times g(\textbf_1, \textbf_2, \textbf_1) + adf \times g(\textbf_1, \textbf_2, \textbf_2) + bce \times g(\textbf_2, \textbf_1, \textbf_1) + bcf \times g(\textbf_2, \textbf_1, \textbf_2) \\ &+ bde \times g(\textbf_2, \textbf_2, \textbf_1) + bdf \times g(\textbf_2, \textbf_2, \textbf_2). \end$

Relation to tensor products

There is a natural one-to-one correspondence between multilinear maps

:$f\colon V_1 \times \cdots \times V_n \to W\text$

and linear maps

:$F\colon V_1 \otimes \cdots \otimes V_n \to W\text$

where $V_1 \otimes \cdots \otimes V_n\!$ denotes the tensor product of $V_1,\ldots,V_n$. The relation between the functions $f\!$ and $F\!$ is given by the formula

:$f(v_1,\ldots,v_n)=F(v_1\otimes \cdots \otimes v_n).$

Multilinear functions on ''n''×''n'' matrices

One can consider multilinear functions, on an matrix over a commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function can be written as

:$D(A) = D(a_,\ldots,a_),$

satisfying

:$D(a_,\ldots,c a_ + a_',\ldots,a_) = c D(a_,\ldots,a_,\ldots,a_) + D(a_,\ldots,a_',\ldots,a_).$

If we let $\hat_j$ represent the th row of the identity matrix, we can express each row as the sum

:$a_ = \sum_^n A(i,j)\hat_.$

Using the multilinearity of we rewrite as

:$D(A) = D\left(\sum_^n A(1,j)\hat_, a_2, \ldots, a_n\right) = \sum_^n A(1,j) D(\hat_,a_2,\ldots,a_n).$

Continuing this substitution for each we get, for ,

:$D(A) = \sum_ \ldots \sum_ \ldots \sum_ A(1,k_)A(2,k_)\dots A(n,k_) D(\hat_,\dots,\hat_).$

Therefore, is uniquely determined by how operates on $\hat_,\dots,\hat_$.

Example

In the case of 2×2 matrices we get

:$D(A) = A_A_D(\hat_1,\hat_1) + A_A_D(\hat_1,\hat_2) + A_A_D(\hat_2,\hat_1) + A_A_D(\hat_2,\hat_2) \,$

Where \hat_1 = ,0/math> and \hat_2 = ,1/math>. If we restrict $D$ to be an alternating function then $D(\hat_1,\hat_1) = D(\hat_2,\hat_2) = 0$ and $D(\hat_2,\hat_1) = -D(\hat_1,\hat_2) = -D(I)$. Letting $D(I) = 1$ we get the determinant function on 2×2 matrices:

:$D(A) = A_A_ - A_A_{2,1} .$

Properties

* A multilinear map has a value of zero whenever one of its arguments is zero.

See also

* Algebraic form
* Multilinear form
* Homogeneous polynomial
* Homogeneous function
* Tensors

References

Multilinear algebra